Iwasawa theory of elliptic curves and modular forms

In this talk, we will give an overview of Iwasawa theory starting with the first case of class groups and then moving on to elliptic curves. In this latter case, the Iwasawa theory of an elliptic curve depends greatly on whether one is working with a prime of ordinary or supersingular reduction. The structure of the ordinary case has been (at least conjecturally) well understood since the 70's while the supersingular case has remained more mysterious. We will discuss this distinction and some recent developments in the supersingular case.